Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(B A\)

Step-by-Step Solution

Answer

Matrix BA is \(\begin{bmatrix} 2 & 24 & -4.5 \\ 12 & 32 & -9 \\ -8 & 64 & 61 \end{bmatrix}\).

Step 1: Verify Matrix Dimensions

Check the dimensions of matrices B and A to determine if the multiplication is possible. Matrix B is a 3x3 matrix and Matrix A is also a 3x3 matrix. Since the number of columns in matrix B (3) matches the number of rows in matrix A (3), matrix multiplication is possible.

Step 2: Multiply Matrices BA

Perform the matrix multiplication for BA by taking the dot product of rows of B with columns of A. Calculate each element of the resulting matrix: - Element (1,1) is calculated as: \((0.5 \times -2) + (3 \times 1) + (0 \times 0.5) = -1 + 3 + 0 = 2\)- Element (1,2) is calculated as: \((0.5 \times 0) + (3 \times 8) + (0 \times 4) = 0 + 24 + 0 = 24\)- Element (1,3) is calculated as: \((0.5 \times 9) + (3 \times -3) + (0 \times 5) = 4.5 - 9 + 0 = -4.5\)- Element (2,1) is calculated as: \((-4 \times -2) + (1 \times 1) + (6 \times 0.5) = 8 + 1 + 3 = 12\)- Element (2,2) is calculated as: \((-4 \times 0) + (1 \times 8) + (6 \times 4) = 0 + 8 + 24 = 32\)- Element (2,3) is calculated as: \((-4 \times 9) + (1 \times -3) + (6 \times 5) = -36 - 3 + 30 = -9\)- Element (3,1) is calculated as: \((8 \times -2) + (7 \times 1) + (2 \times 0.5) = -16 + 7 + 1 = -8\)- Element (3,2) is calculated as: \((8 \times 0) + (7 \times 8) + (2 \times 4) = 0 + 56 + 8 = 64\)- Element (3,3) is calculated as: \((8 \times 9) + (7 \times -3) + (2 \times 5) = 72 - 21 + 10 = 61\)

Step 3: Construct the Result Matrix

Using the calculated values from Step 2, construct the resulting matrix:\[BA = \begin{bmatrix}2 & 24 & -4.5 \12 & 32 & -9 \-8 & 64 & 61\end{bmatrix}\]

Step 4: Verify with a Calculator

To ensure accuracy, use a calculator or matrix computation tool to verify the elements of the result matrix.

Key Concepts

Matrix Operations

Matrix operations are fundamental aspects of linear algebra that involve operations like addition, subtraction, and multiplication of matrices. In this particular exercise, we focus on matrix multiplication. Unlike addition and subtraction, which require matrices to have the same dimensions, multiplication involves a more complex operation where not all matrices can be multiplied together. This is because multiplication depends on the internal structure of the matrices, particularly their dimensions. Understanding matrix operations helps solve a wide array of mathematical and real-world problems.

When multiplying two matrices, the dot product is used to calculate each element of the resulting matrix. This requires pairing elements from the rows of the first matrix with the columns of the second. If you learn these steps, you can better understand how complex matrices alter data through transformations.

Dot Product

To perform matrix multiplication, we use the dot product method. The dot product takes two equal-length sequences of numbers and multiplies corresponding elements and then sums them.

• The dot product is integral to determining the elements of the resulting matrix.
• For example, when finding a single element in a resulting matrix, select a row from the first matrix and a column from the second matrix. Multiply each corresponding element and add them up.

In the exercise, we calculated all the elements in the result matrix using the dot product. For each element, such as the first element (1,1), we selected the first row of matrix B and the first column of matrix A, performed element-wise multiplication, and added up the resulting products: \((0.5 imes -2) + (3 imes 1) + (0 imes 0.5) = 2\).
Understanding the dot product is a step toward grasping matrix transformations utilized in various applications such as computer graphics and statistics.

Matrix Dimensions

Matrix dimensions are crucial in matrix operations, particularly in multiplication. The term dimensions refer to the number of rows and columns in a matrix, written as m x n, where m is the number of rows, and n is the number of columns. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Only then can the dot product be performed for each element pair.

For example, in our exercise:

• Matrix B is 3x3, meaning 3 rows and 3 columns.
• Matrix A is also 3x3, meaning 3 rows and 3 columns.

Since the inner dimensions match (3 from B and 3 from A), matrix multiplication is possible, resulting in a matrix with the dimensions of the outer numbers of the original matrices, which is also 3x3. Checking and understanding matrix dimensions before attempting multiplication can save time and reduce errors in calculations.

Algebra

The concept of algebra encompasses a wide range of topics, including the manipulation of symbols and expressions to solve equations. In the realm of matrices, algebraic principles help define and perform operations that have concrete applications in various fields like engineering, computer science, and economics.

Matrix multiplication itself is grounded in algebraic rules. The distributive property, for instance, is used implicitly when calculating the dot product for each element in the resulting matrix. These operations resemble solving linear equations, where each step requires attention to order and operation, ensuring accuracy. Mastering algebra through matrices offers a gateway to understanding more abstract mathematical concepts and applying them effectively in practical situations.